Monster Lie algebra

Description

Write the j-invariant as $j(q) - 744 = \sum_{n \ge -1} c(n)\, q^n = q^{-1} + 196884\, q + 21493760\, q^2 + \cdots$, so $c(-1) = 1$, $c(0) = 0$ and $c(1) = 196884$. The monster Lie algebra $\mathfrak{m}$ is a generalized Kac–Moody algebra graded by $\Z^2$ with these properties:

1. its Cartan subalgebra is the degree $(0, 0)$ part, of dimension $2$;
2. for $(m, n) \neq (0, 0)$, the part of degree $(m, n)$ has dimension $c(mn)$;
3. it has one real simple root, of degree $(1, -1)$, and for each $n \ge 1$ it has $c(n)$ imaginary simple roots of degree $(1, n)$.

Its denominator identity is the product formula $$j(p) - j(q) = p^{-1} \prod_{m > 0,\ n \in \Z} \left(1 - p^m q^n\right)^{c(mn)}.$$ Borcherds constructed $\mathfrak{m}$ from the moonshine module $V^\natural$, a vertex operator algebra on which the Monster group acts. The construction also uses the vertex algebra of the even unimodular Lorentzian lattice of rank $2$ and the no-ghost theorem from string theory. The Monster group acts on $\mathfrak{m}$ by automorphisms preserving the grading.